Overview
After O26 introduced a representation-theoretic interpretation of the pair observable and O28 measured an effective rank \(r_{\mathrm{eff}} = 3\) with eigenvalue structure \([1 : \tfrac{1}{2} : \tfrac{1}{2}]\), a question remained: the naive target for the spin-\(\tfrac{1}{2}\) candidate was \(d_\rho^2 = 4\).
The central result of O29 is: \(r_{\mathrm{eff}} = 3\) is the invariant of the irreducible adjoint (spin-\(1\)) carrier \(H_{\mathrm{eff}} \simeq \mathrm{Sym}^2(V_\rho)\), not a symmetric-rank inversion to a two-dimensional subspace.
On the existing checkpoints the admissible trajectory spans all three complex dimensions of \(H_{\mathrm{eff}}\), and \(r_{\mathrm{eff}} = 3\) is a hard \(6 \to 3\) collapse of the Veronese (squaring) image: the three constraint forms span \(\mathfrak{so}(3)\) and the action on \(H_{\mathrm{eff}} = \mathbb{C}^3\) is irreducible (Schur commutant one-dimensional).
The anti-linear Born–Infeld parity makes the conjugate-pair target \(d_\rho^2 = 4\) inaccessible by construction (restored only by the locking-broken control). The spin-\(\tfrac{1}{2}\) space \(V_\rho\) (\(d_\rho = 2\)) is recovered as the Veronese square-root structure of the data.
Main contributions
- Symmetric outer products: conjugate-pair outer products satisfy \(M_j = w_j w_j^\top\) (complex symmetric), under anti-linear Born–Infeld parity.
- Hard \(6 \to 3\) Veronese collapse: the symmetric squares satisfy exactly three quadratic relations; the data lie on a Veronese cone.
- \(\mathfrak{so}(3)\) structure: the three constraint forms are traceless and their commutators span \(\mathfrak{so}(3)\).
- Irreducible adjoint carrier: the Schur commutant is one-dimensional, so \(H_{\mathrm{eff}} \simeq \mathfrak{su}(2) \cong \mathrm{Sym}^2(V_\rho)\) is the spin-\(1\) representation.
- Inaccessibility of \(d_\rho^2 = 4\): the conjugate-pair target is unreachable by construction; \(V_\rho\) (\(d_\rho = 2\)) is the Veronese square-root coordinate, not the rank measured by Test 4.
Interpretation
O29 fixes the interpretation of the rank observable.
- What Test 4 measures: the irreducible adjoint (spin-\(1\)) carrier \(H_{\mathrm{eff}} \simeq \mathrm{Sym}^2(V_\rho)\)
- What it does not measure: the spin-\(\tfrac{1}{2}\) target \(V_\rho\), recovered only through the Veronese square-root structure
The key conceptual point is: \(r_{\mathrm{eff}} = 3\) counts the dimension of the spin-\(1\) carrier \(H_{\mathrm{eff}} = \mathrm{Sym}^2(V_\rho)\), not a symmetric rank of a two-dimensional subspace.
Relation to the Cosmochrony programme
O29 follows O27 by fixing the carrier of the representation identification, consistent with the \(H_{\mathrm{eff}} \simeq \mathrm{Sym}^2(V_\rho)\) reading of Q7.
The sequence now reads: O16–O23 (pair structure and admissibility), O24 (rank stability), O25 (numerical validation), O26 (quadratic completion), O27 (SU(2) rigidity), O28 (rank observation), O29 (carrier identification), O30 (analytical \([1 : \tfrac{1}{2} : \tfrac{1}{2}]\) eigenvalue ratio).
It provides the missing link between numerical observation and the adjoint \(\mathfrak{su}(2)\) carrier.
Current result and open directions
- Full observable: design a protocol probing the full \(d_\rho^2\) space (the locking-broken control reaches it).
- Analytical dimension: derive the carrier dimension \(\dim H_{\mathrm{eff}} = 3\) and the Veronese collapse for all primes.
- Universality: extend the carrier diagnostics to larger primes.
- Eigenvalue structure: the ratio \([1 : \tfrac{1}{2} : \tfrac{1}{2}]\) is derived analytically in O30.
Reference
Jérôme Beau. Carrier Identification via the Veronese Collapse: Effective Dimension of the Admissible Covariance in End(H_eff).